Suppose that a function f is differentiable at x0 and that f'(x0)>0. Prove that there exists an open interval containing x0 such that if x1 and x2 are any two points in this interval with x1 < x0 < x2 then f(x1) < f(x0) < f(x2). How do I establish an open interval? Do I need the "epsilon delta" definition of a two sided limit? I am studing maths on my own so please help with this easy question. I know, that if a function f is differentiable at x0, then f is continuous at x0 and as f'(x0) is positive then f is an increasing function at x=x0.